A candle flame 3 cm high is placed at a distance of 3 m from a wall. How far from the wall must a concave mirror be placed so that it forms an image of the flame 9 cm high on the wall?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 450 cm
An image formed on a wall is a real image, so the concave mirror forms a real, inverted image of the flame. The magnification of a mirror is \( m=h'/h \), and also \( m=-v/u \).
\( \begin{aligned} m &= \frac{h'}{h} \\ &= \frac{-9\,\mathrm{cm}}{3\,\mathrm{cm}} \\ &= -3 \end{aligned} \)
\( -v/u=-3 \)
\( \Rightarrow\; v=3u \)
Both the flame and its image are in front of the mirror. The image on the wall is three times as far from the mirror as the flame. The flame is 300 cm from the wall, so the two distances differ by 300 cm:
\( |v|-|u|=300\,\mathrm{cm} \)
\( \begin{aligned} 3|u|-|u| &= 300\,\mathrm{cm} \\ |u| &= 150\,\mathrm{cm} \end{aligned} \)
\( \begin{aligned} |v| &= 3|u| \\ &= 3\times150\,\mathrm{cm} \\ &= 450\,\mathrm{cm} \end{aligned} \)
In the Cartesian sign convention \( u=-150\,\mathrm{cm} \) and \( v=-450\,\mathrm{cm} \). The image is on the wall, so the mirror is placed 450 cm from the wall.